Definition:Jordan Arc

Definition

Let $\left({x_1, y_1}\right), \left({x_2, y_2}\right) \in \R^2$.

Let $f: \left[{0 \,.\,.\, 1}\right] \to \R^2$ be a path from $\left({x_1, y_1}\right)$ to $\left({x_2, y_2}\right)$.

Then $f$ is a Jordan arc if and only if $f$ is an injection, except that we allow the possibility $\left({x_1, y_1}\right) = \left({x_2, y_2}\right)$.

Also defined as

In many sources, a Jordan arc $f$ is defined as a path that is an injection, so the initial point of $f$ is different from the final point of $f$.

That is, $f$ is a homeomorphism of the closed unit interval $\left[{0 \,.\,.\, 1}\right]$.

Also see

• Definition:Jordan Curve

Source of Name

This entry was named for Marie Ennemond Camille Jordan.